The 34-year-old Quesne made six birdies in Shanghai to put himself in sight of his third European Tour title and first since 2013.
His eight-under total left him one clear of Levy, who looked set to grab the halfway lead before finding water with his approach at the par-four 16th and double bogeying the hole.
Levy, also 34 and looking for his third European title, followed with another bogey at the par-three 17th but birdied his last to move within one on seven under after a 68, alongside American Peter Uihlein (67).
Thailand's Kiradech Aphibarnrat, who won last week's European Tour title in Shenzhen, moved into contention for another victory after firing a 68 early on Friday to move into a tie for fourth.
Italy's Matteo Manassero matched the Thai's mark with Wu Ashun (66) and Li Hao-tong (68) leading the local charge to join the pair at five-under.
(Writing by Patrick Johnston in Singapore, editing by Ed Osmond)
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